| “Hom和Johnson教授合著的《矩阵分析 卷1》和《矩阵分析 卷2》是不可或缺的。首先，深入地探讨了每个论题；其次，书中涵盖了一些高级认题；第三，精心组织，内容广泛，是一本标准的教科书。” ——国际线性代数学报 “毫无疑问，《矩阵分析 卷1》和《矩阵分析 卷2》是当今矩阵理论文献中非常重要的而且无可比拟的著作。” ——美国数学学会会刊 本书是继《矩阵分析 卷1》之后推出的矩阵领域又一经典之作，详尽讨论了卷1未能包括但又具有极高应用价值的论题。 书中包含大量矩阵理论和线性代数方面的经典定理和推论，并给出了严格的证明。很多定理、推论、论题等都是本书所独有的，再加上作者精心组织，言简意赅的表述，造就了本书在矩阵领域不可比拟、独一无二的地位。 |
| Roger A.Horn 线性代数和矩阵理论领域国际知名权威。1967年获得斯坦福大学数学博士，1972-1979年任约翰·霍普金斯大学数学系系主任，现为犹他大学教授。曾担任AMERICAN MATHE-MATICAL MONTHLY编辑。
Charles R.Johnson线性代数和矩阵理论领域国际知名权威。现为威廉玛丽学院教授。Johnson在学术界十分活跃，发表论文近300篇，担任过多个主要矩阵分析类杂志的编辑和两份SIAM杂志的主编。由于他在数学科学领域作出了杰出贡献而被授予华盛顿科学学会奖。 |
| Chapter 1 The field of values 1 1.0 Introduction 1 1.1 Definitions 5 1.2 Basic properties of the field of values 8 1.3 Convexity 17 1.4 Axiomatization 28 1.5 Location of the field of values 30 1.6 Geometry 48 1.7 Products of matrices 65 1.8 Generalizations of the field of values 77 Chapter 2 Stable matrices and inertia 89 2.0 Motivation 89 2.1 Definitions and elementary observations 91 2.2 Lyapunov's theorem 95 2.3 The Routh-Hurwitz conditions 101 2.4 Generalizations of Lyapunov's theorem 102 2.5 M-matrices, P-matrices, and related topics 112 Chapter 3 Singular value inequalities 134 3.0 Introduction and historical remarks 134 3.1 The singular value decomposition 144 3.2 Weak majorization and doubly substochastic matrices 163 3.3 Basic inequalities for singular values and eigenvalues 170 3.4 Sums of singular values: the Ky Fan k-norms 195 3.5 Singular values and unitarily invariant norms 203 3.6 Sufficiency of Weyl's product inequalities 217 3.7 Inclusion intervals for singular values 223 3.8 Singular value weak majorization for bilinear products 231 Chapter 4 Matrix equations and the Kronecker product 239 4.0 Motivation 239 4.1 Matrix equations 241 4.2 The Kronecker product 242 4.3 Linear matrix equations and Kronecker products 254 4.4 Kronecker sums and the equation AX + XB = C 268 4.5 Additive and multiplicative commutators and linear preservers 288 Chapter 5 The Hadamard product 298 5.0 Introduction 298 5.1 Some basic observations 304 5.2 The Schur product theorem 308 5.3 Generalizations of the Schur product theorem 312 5.4 The matrices A o (A-1) T and A o A-1 322 5.5 Inequalities for Hadamard products of general matrices: an overview 332 5.6 Singular values of a Hadamard product: a fundamental inequality 349 5.7 Hadamard products involving nonnegative matrices and M-matrices 356 Chapter 6 Matrices and functions 382 6.0 Introduction 382 6.1 Polynomial matrix functions and interpolation 383 6.2 Nonpolynomial matrix functions 407 6.3 Hadamard matrix functions 449 6.4 Square roots, logarithms, nonlinear matrix equations 459 6.5 Matrices of functions 490 6.6 A chain rule for functions of a matrix 520 Hints for problems 561 References 572 Notation 575 Index 580 |
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